| L(s) = 1 | − 2-s + 4-s + 3.83·5-s + 5.08·7-s − 8-s − 3.83·10-s + 4.51·11-s − 3.32·13-s − 5.08·14-s + 16-s − 0.154·17-s + 5.25·19-s + 3.83·20-s − 4.51·22-s + 9.71·25-s + 3.32·26-s + 5.08·28-s + 5.23·29-s + 1.26·31-s − 32-s + 0.154·34-s + 19.4·35-s − 4.45·37-s − 5.25·38-s − 3.83·40-s − 2.21·41-s − 2.63·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.71·5-s + 1.92·7-s − 0.353·8-s − 1.21·10-s + 1.36·11-s − 0.923·13-s − 1.35·14-s + 0.250·16-s − 0.0373·17-s + 1.20·19-s + 0.857·20-s − 0.963·22-s + 1.94·25-s + 0.652·26-s + 0.960·28-s + 0.971·29-s + 0.227·31-s − 0.176·32-s + 0.0264·34-s + 3.29·35-s − 0.732·37-s − 0.851·38-s − 0.606·40-s − 0.345·41-s − 0.402·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.340470010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.340470010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 - 5.08T + 7T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 + 3.32T + 13T^{2} \) |
| 17 | \( 1 + 0.154T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + 4.45T + 37T^{2} \) |
| 41 | \( 1 + 2.21T + 41T^{2} \) |
| 43 | \( 1 + 2.63T + 43T^{2} \) |
| 47 | \( 1 + 1.10T + 47T^{2} \) |
| 53 | \( 1 - 4.28T + 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 6.20T + 73T^{2} \) |
| 79 | \( 1 + 2.91T + 79T^{2} \) |
| 83 | \( 1 - 2.33T + 83T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64328911895616289006730088631, −7.14544743203731124580487783964, −6.34962677267248951441453520139, −5.69542365229845389609970513732, −4.98929355039536539264101165534, −4.50718884552384561290781410266, −3.14134916994514353902997567189, −2.22859010971834066878272033974, −1.56612883311143118242682906511, −1.13855406959330511262619816434,
1.13855406959330511262619816434, 1.56612883311143118242682906511, 2.22859010971834066878272033974, 3.14134916994514353902997567189, 4.50718884552384561290781410266, 4.98929355039536539264101165534, 5.69542365229845389609970513732, 6.34962677267248951441453520139, 7.14544743203731124580487783964, 7.64328911895616289006730088631
